White Dwarfs and Neutron Stars

This is a brief series of articles on stellar evolution, neutron stars, and black holes. This series consists of 3 parts:

  1. Stellar Evolution - Part I of the article
  2. White Dwarfs and Neutron Stars - Part II of the article: this page
  3. Black Holes - Part III of the article

The series begins with a brief introduction to star formation, covering the birth, lifetime and death of a star. The second part describes neutron stars, including special types of neutron stars, such as pulsars and magnetars. The third part describes black holes.

In the previous section, we covered stellar evolution, covering the history of a star from its birth to its death. This section and the next describe star remnants, the objects left behind when a star dies. As described in the previous section, what remains when the star dies depends upon how big the star was, and how violently it died. Stars that are smaller than about 3 S (S represents the mass of the Sun, around 1.9891 x 1030 kilograms) produce white dwarfs. Stars that are between 3 S and about 8-11 S produce neutron stars. Stars that are even larger produce black holes. Remember, the mass of the neutron star or black hole doesn't correlate that well with the mass of the original star that produced it. While the Chandrashekhar Limit for neutron stars and the Tolman-Oppenheimer-Volkoff limit for black holes are much lower (1.4 S and about 2-3 S respectively), these are the masses of the star's core after implosion. Since stars lose most of their mass during implosion, a star of 8-10 S could leave behind a core of only 2 S.

After a star has burned through all its fuel, it collapses, since no more heat (and therefore no more pressure) is being generated inside it, and there is nothing to oppose gravity. The collapse produces enormous amounts of energy, as the gravitational potential energy of the star gets converted to heat. The outer layers of the star are blown off into space, and a very dense core remains. The nature of this core is described in the next two sections.

White Dwarfs

As the core collapses under gravity, it grows denser and denser. White dwarfs produced even by small stars are very dense objects. White dwarfs discovered so far range between 0.17 to 1.33 S in mass, though the majority fall within the range 0.5 - 0.7 S. However, they are tiny objects, usually about 1- 2 Earth-radii. Obviously an object half the mass of the Sun that is only about the size of the Earth must be very dense. White dwarf densities have been calculated to be around 1 ton per cubic centimeter.

Such densities are not possible for ordinary matter. The densest known element, Osmium, only has a density of 22.6 grams per cubic centimeter. However much you try to squeeze matter into a smaller space, you eventually run up against the fact that you can't bring atoms together any closer than their outer electron clouds will allow. This is not true for white dwarfs. White dwarfs are made of a kind of plasma of free electrons and nuclei. Since this material is not atoms, there is no restriction in regard to moving electrons closer than the electron orbitals in an atom would allow. This is what permits the squeeze.

Initially, it was thought that the high temperature of white dwarf matter kept the material ionized (i.e., separated into unbound electrons and nuclei). Later, it was shown that this state could exist even at absolute zero. This can be understood in terms of the uncertainty principle. With electrons packed together so closely, their positions are highly localized. Therefore, according to the uncertainty principle, their momenta must have high uncertainty. This means that there must be electrons with very high momentum (and therefore high kinetic energy). So this kind of material can be at absolute zero, and still possess high energy.

White dwarf matter is quite compressible. In fact, white dwarfs get smaller as they increase in mass, simply because their material gets more compressed. As you continue to compress white dwarf matter, the electrons simply move even closer together. This increases their localization, and therefore also increases the uncertainty in their momenta - and consequently their kinetic energies. The kinetic energy of particles is the basis of pressure (see kinetic theory of gases), therefore increasing the density of white dwarf matter increases kinetic energy of electrons, which increases pressure. This pressure is known as the electron degeneracy pressure, and this is what supports the white dwarf against gravitational collapse.

In 1930, the astrophysicist Subramanyan Chandrashekhar calculated that the electron degeneracy pressure was sufficient to maintain a non-rotating white dwarf of about 1.4 S. If a white dwarf exceeded this mass, the electron degeneracy pressure would not be enough to prevent gravitational collapse. This mass of 1.4 S is known as the Chandrashekhar Limit. Non-rotating white dwarfs with masses greater than 1.4 S will collapse into some denser form of matter.

If the white dwarf is rotating, as most are, we need to take into account the centrifugal pseudo-force, which counteracts gravity. Rotating white dwarfs can therefore be somewhat larger than the Chandrashekhar Limit. For uniformly rotating white dwarfs, the difference is not great.

Since white dwarfs are formed due to gravitational collapse of stars, they start their lives very hot, due to the massive release of gravitational potential energy during the collapse. White dwarfs have been found with surface temperatures as hot as 150,000 °K. Their small size (and therefore small surface area) means that they lose heat very slowly. In fact, the slow cooling of white dwarfs can be used to date them, with the coldest known white dwarf (at a surface temperature of 3,900 °K) being about 8 billion years old. This helps date the galactic disk of the Milky Way to at least that age.

Eventually, white dwarfs cool down to reach thermal equilibrium with their surroundings, become black dwarfs, and cease to radiate. It is believed that no black dwarfs exist in the universe, since the universe is not yet old enough.

Neutron Stars

White dwarfs larger than 1.4 S collapse into neutron stars. There is an upper limit to neutron star size, known as the Tolman-Oppenheimer-Volkoff (TOV) limit, which will be explained later. Because of this limit, it is believed that there are no neutron stars massing more than 2 - 3 S.

Neutron stars are much denser than white dwarfs. The equations of state for neutron stars have not been completely worked out. White dwarf matter is thought to behave like a degenerate gas, whose properties can be worked out with special relativity. However, for a neutron star, general relativity cannot be ignored. Several equations of state for neutron stars have been proposed. A typical neutron star of mass 1.5 S would have a radius anywhere between 10.7 - 15.1 km, depending on which equation of state is used.

Artist's Impression of a Neutron Star, showing magnetic lines of force. From NASA.

Neutron star matter is also thought to be fairly compressible. The density of neutron star matter varies between < 1000 tons per cubic centimeter at the surface, to as much as 800 million tons per cubic centimeter at the center. These numbers are approximate, they vary somewhat depending upon the equation of state used, but they are in the ballpark. If you compare this to the density of the atomic nucleus - about 300 million tons per cubic centimeter - you can see that the density at the center of a neutron star can be much higher than that of the atomic nucleus.

Neutron stars are formed at very high temperatures. The initial supernova and collapse can produce neutron stars as hot as 100 billion to 1 trillion °K, but because of the huge neutrino flux, it quickly falls within a few years to about a million °K. These high temperatures indicate that neutron stars emit most of their energy as x-rays. Neutron stars spin very rapidly, due to the conservation of angular momentum. The stars they are created from have slow spins, but the rapid shrinkage of a stellar sized mass into a radius of only 10-20 km magnifies the spin tremendously. A brand new neutron star can rotate at several times per second. The spin is very gradually lost as the rotating magnetic field loses energy. However, neutron stars can speed up, if they orbit another star and are able to accrete matter from it.

Rotating neutron stars sometimes emit beams of electromagnetic radiation from their magnetic poles. The mechanism by which these beams are produced is not well understood, but it's possible that it has to do with particle acceleration within their very strong magnetic fields (the magnetic poles do not coincide with the axis of rotation). Because of the rotation, we observe these beams in pulses, once per rotation. Such radio wave emitting neutron stars are called pulsars.

Because of these pulses, the rotation rates of many neutron stars can be easily observed. Neutron stars with spins of up to 716 revolutions per second are known. The slowest measured rotation has been about 1 rotation every 30 seconds. An isolated observation (not confirmed yet) shows a millisecond pulsar with a calculated rotational speed of 1122 rotations per second. Such a high speed does not fit in with current models, which predict that at speeds >1000 rotations per second, they should lose energy through gravitational radiation faster than they can speed up through the accretion process. The currently accepted upper limit structurally is about 1500 rotations per second. Any higher would tear the neutron star apart through centrifugal forces.

Neutron stars have a layered structure. The top 1 meter or so (the "atmosphere") consists of a mix of atomic nuclei and electrons. If the surface of the neutron star is hotter than about a million °K, this "atmosphere" would be in liquid form, while if it's cooler than that, it would be solid. Below that is a solid crust, about a kilometer thick. This crust is very hard and very smooth. Gravity would probably prevent any irregularities larger than half a centimeter. Below the crust, all the way to the center, is a liquid interior.

As we go deeper into the neutron star, the material consists of nuclei and electrons, with the nuclei containing an increasing number of neutrons. Such nuclei could not exist normally, they would decay, but the extreme pressure keeps them stable. Going deeper, the neutron drip limit is reached, at which point neutrons start leaking out of the nuclei. Here, neutron star matter is a mix of nuclei, free electrons, and free protons. Going deeper still, the size of the nuclei decreases, and the number of free neutrons increases. Hypothetically, at the core there would be no more free nuclei.

[CLICK HERE FOR LARGE VERSION] Diagram of the interior of a neutron star.

The exact nature of the material at the core is unknown. Some people call it "neutron degenerate matter" (scifi calls it "neutronium"). According to Wikipedia:

"Neutron star core material could be a superfluid mixture of neutrons with a few protons and electrons, or it could incorporate high-energy particles like pions and kaons in addition to neutrons, or it could be composed of strange matter incorporating quarks heavier than up and down quarks, or it could be quark matter not bound into hadrons. However, so far, observations have neither indicated nor ruled out such exotic states of matter."

In other words, nobody has a clue.

Neutron stars have very powerful magnetic fields. As the original star collapses into a neutron star, the magnetic field increases exponentially (each halving of radius increases the magnetic field 4x). The typical magnetic field of a neutron star is probably around 100 million Tesla. The Earth's magnetic field is only 30-60 micro Tesla, so the neutron star is over a trillion times more powerful. However, since the interior of neutron stars is a conducting liquid, the dynamo mechanism can increase this field tremendously for brief durations. Fields of up to 200 billion Tesla have been calculated. Such neutron stars are called magnetars. It's been calculated that about 1 in 10 supernova explosions result in a magnetar, rather than the usual neutron star. About 15 magnetars are known so far, with the nearest being about 13,000 light years away.

The most powerful magnetar so far known is SGR 1806-20, about 50,000 light years from Earth, in the constellation of Sagittarius. It's less than 10 kilometers in radius, with a rotation period of once per 7.5 seconds. It was discovered in 2004, when on Dec 27, a gamma ray burst reached the Earth. If the output had been visible light instead of gamma rays, the burst would have been brighter than the full moon, when viewed from Earth. It remains the brightest known extra-solar event, and the second most powerful explosion known to have been viewed by humans (the first would be the SN 1604 supernova, viewed by Johannes Kepler). The SGR 1806-20 burst released more energy in 1/10 of a second than the Sun has released in 250,000 years. If this magnetar had been within 10 light years of Earth, it would have destroyed the ozone layer, producing a massive extinction event. The blast would pack the energy of a 12 kiloton nuclear explosion at 7.5 kilometers.

The magnetic field of SGR 1806-20 is estimated to be around 200 billion Tesla. A magnetar with such a powerful field could wipe a credit card at distance half that to the moon. At 1000 kilometers, it could tear apart flesh, due to the diamagnetism of water. At the surface, the magnetic field would be so strong, that its energy density would be around 1.6x1028 Joules per cubic meter. To give a rough idea of what this means, each cubic centimeter of space around the surface of the magnetar would contain the energy (magnetic only, not counting the gravitational potential energy) equivalent of about 250 million Hiroshima-type nuclear bombs. Link to calculations here.

Such strong fields cannot be maintained for long. It's estimated that the "active" life of a magnetar is only about 10,000 years, after which it becomes an ordinary neutron star. Given the number of observable magnetars today, it's been calculated that there may be as many as 30 million or more inactive magnetars in the Milky Way alone.

Quakes on the surface of magnetars can cause deformation of the massive magnetic fields, resulting in very powerful gamma ray bursts. The kind of bursts produced by magnetars are called soft gamma repeaters, indicating that they are both periodic, and less energetic than other types of gamma ray bursts. They are still powerful enough to force the shutdown of sensors in some satellites and spacecraft, even from distances of 20,000 light years or more. Gamma ray bursts produced by magnetars are so far the most powerful extra solar bursts to have been experienced on Earth.

Note however, that even more powerful gamma ray bursts have been observed in distant galaxies, but because they are billions of light years away, the effects on Earth were minor. These very powerful bursts are the most luminous events in the universe since the big bang. Most typically last a few seconds, outputting as much energy in those few seconds as the Sun will in its entire lifetime, or about 1/2000 the mass equivalent of the Sun. They are usually classified as either long bursts (> 2 seconds) or short bursts. The short bursts may be due to the collision of neutron stars. Long bursts are likely produced in supernova explosions. Such explosions can often direct the energy in a relatively narrow beam, instead of dissipating it spherically. For this reason, the intensity of gamma ray bursts appears much magnified, if the Earth is caught within the beam. We observe on average about 1 gamma ray burst per day. Since gamma ray bursts are powerful enough to be seen from the farthest reaches of the universe, the frequency of their production must be very low. For a galaxy the size of the Milky Way, they may happen once every 100,000 - million years. However, since the energy is mostly beamed at a narrow angle, the likelihood of the Earth getting caught in such a beam is even smaller, perhaps 1 in a billion or less. However, if such a burst were to occur within our own galaxy, and if the Earth got caught in the beam, it could lead to a major extinction of life of Earth. Some people have hypothesized that the Ordovician - Silurian extinction event about 450 million years ago was caused by such a gamma ray burst in our own galaxy. Evidence for this hypothesis is poor.

At present, there are about 2000 known neutron stars, mostly in our galaxy, but also some in the Magellanic Clouds. The closest we know of is about 200 light years away in the constellation Corona Australis, with a surface temperature of about 600,000 °K. About half the known neutron stars are parts of a binary system, with the companion stars either being ordinary stars, white dwarfs, or other neutron stars. It's possible that other binary systems may exist where the binary companion of the neutron star is a black hole.

Continue to read about black holes in the next section.